ABOARD THE USS THEODORE ROOSEVELT—U.S. Defense Secretary Ash Carter sent a blunt message to China on Thursday by joining the American aircraft carrier USS Theodore Roosevelt on patrol in the sensitive South China Sea: that the U.S. won’t accept Chinese domination of the contested region.

Mr. Carter appeared on the Theodore Roosevelt as it navigated some 200 miles from the southern tip of the disputed Spratly Islands. He sought to balance the American desire to maintain an open dialogue with the Chinese with the projection...